Persistent Cup-Length

Cohomological ideas have recently been injected into persistent homology and have been utilized for both enriching and accelerating the calculation of persistence diagrams. For instance, the software Ripser fundamentally exploits the computational advantages offered by cohomological ideas. The cup product operation which is available at cohomology level gives rise to a graded ring structure which extends the natural vector space structure and is therefore able to extract and encode additional rich information. The maximum number of cocycles having non-zero cup product yields an invariant, the Cup-Length, which is efficient at discriminating spaces. In this paper, we lift the cup-length into the Persistent Cup-Length invariant for the purpose of extracting non-trivial information about the evolution of the cohomology ring structure across a filtration. We show that the Persistent Cup-Length can be computed from a family of representative cocycles and devise a polynomial time algorithm for the computation of the Persistent Cup-Length invariant. We furthermore show that this invariant is stable under suitable interleaving-type distances. Along the way, we identify an invariant which we call the Cup-Length Diagram, which is stronger than persistent cup-length but can still be computed efficiently. In addition, by considering the $\ell$-fold product of persistent cohomology rings, we identify certain persistence modules, which are also stable and can be used to evaluate the persistent cup-length.